Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems

and Applied Analysis 3 2. Preconditioned Explicit Decoupled Group SOR (EDG SOR) For convenience, we will now briefly explain some of the definitions used in this paper. Definition 2.1 see 15 . A matrix A of order n has property A if there exists two disjoint subsets S and T of W {1, 2, . . . , n} such that if i / j and if either aij / 0 and aij / 0, then i ∈ S and j ∈ T or else i ∈ T and j ∈ S. Definition 2.2 see 3 . An ordered grouping π of W {1, 2, . . . , n} is a subdivision of W into disjoint subsets R1, R2, . . . , Rq such that R1 R2 · · · Rq W . Given a matrix A and an ordered grouping π , we define the submatrices Am,n for m,n 1, 2, . . . q as follows: Am,n is formed from A deleting all rows except those corresponding to Rm and all columns except those corresponding to Rn. Definition 2.3 see 3 . Let π be an ordered grouping with q groups. A matrix A has Property A π if the q × q matrix Z zr,s defined by zr,s {0 if Ar,s 0 or 1 if Ar,s / 0} has Property A. Definition 2.4 see 15 . A matrix A of order n is consistently ordered if for some t there exist disjoint subsets S1, S2, . . . , St of W {1, 2, . . . , n} such that ∑t k 1 Sk W and such that if i and j are associated, then j ∈ Sk 1 if j > i and j ∈ Sk−1 if j < i, where Sk is the subset containing i. Note that a matrix A is a π-consistently ordered matrix if the matrix Z in Definition 2.3 is consistently ordered. From the discretisation of the EDG finite-difference formula in solving the Poisson equation, the linear system